Applications of the Particle in a Box

Chem 3240 · Lecture 3.4

Davit Potoyan

Why a Toy Model Matters

The particle in a box is exactly solvable.

It also predicts real chemistry: light absorbed by dye-like molecules.

Key idea: delocalized \(\pi\)-electrons behave like particles trapped in a 1D box.

\(\pi\)-Electrons as Confined Particles

Conjugated chains: alternating single and double bonds.

\(\pi\)-electrons are delocalized over the chain.

Box length \(L\) = length of the conjugated region.

The Model System: Butadiene

Butadiene \(\text{C}_4\text{H}_6\): four carbons, 4 \(\pi\)-electrons.

C–C single bond: 1.54 Å

C=C double bond: 1.34 Å

Step 1: Box Length

Sum the bond lengths across the conjugated chain.

\[ L = 1.54\, \text{Å} + 1.34\, \text{Å} + 1.54\, \text{Å} = 4.42\, \text{Å} \]

Each C contributes one \(\pi\)-electron \(\rightarrow\) 4 electrons total.

Step 2: Energy Levels

1D particle-in-a-box energies:

\[ E_n = \frac{n^2 h^2}{8mL^2} \]

\(n = 1, 2, 3, \dots\), \(\;m\) = electron mass, \(\;L\) = box length.

Step 3: Fill and Excite

4 electrons, 2 per level \(\rightarrow\) fill \(n=1\) and \(n=2\).

HOMO \(= n=2\), LUMO \(= n=3\).

Absorption promotes one electron: \(n=2 \rightarrow n=3\).

Step 4: Photon Wavelength

Transition energy: \[ \Delta E = E_{n+1} - E_n \]

Set equal to photon energy: \[ \Delta E = \frac{hc}{\lambda} \]

Solve for \(\lambda\) \(\rightarrow\) the color absorbed.

The Trend: Longer Chain, Redder Light

\(\Delta E \propto \dfrac{1}{L^2}\)

Bigger conjugation \(\rightarrow\) smaller \(\Delta E\) \(\rightarrow\) longer \(\lambda\).

Explains why extended dyes are deeply colored.

Beyond 1D: 2D Box

Aromatic rings and graphene fragments: confine in two directions.

\[ E_{n_x,n_y} = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} \right) \]

Same physics, two quantum numbers.

Takeaway

The particle in a box turns a length and an electron count into a predicted absorption wavelength: longer conjugation gives a smaller gap and redder color.